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Comparison principles for nonlinear operator differential equations in Banach spaces. (English) Zbl 0923.34057
Buslaev, V. (ed.) et al., Differential operators and spectral theory. M. Sh. Birman’s 70th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 189(41), 149-157 (1999).
Existence theorems are obtained for a class of semilinear operator differential equations. For it linear integral inequalities for \(t\mapsto \| Tv\| _{W_p^l (t,t+1)} \), \(t\mapsto \| T_qv\| _{W_p^l (t,t+1)}\), \(t\mapsto \| Tv-T_qv\| _{W_p^l (t,t+1)}\), \(t\mapsto \| Tv-v\| _{W_p^l (t,t+1)}\) are derived where \(T\), \(T_q\) are nonlinear operators connected with the differential equation. Comparing the integral inequalities with the corresponding integral equations, the required estimates and especially norm majorants of the solution to the differential equation by solutions to the integral equation are found if the integral equations are solvable.
For the entire collection see [Zbl 0911.00011].
34G20 Nonlinear differential equations in abstract spaces
47J25 Iterative procedures involving nonlinear operators